Prove that 5√2 is irrational.

1.	Prove that 5√2 is irrational.
Answer» Let 5√2 is a rational number. ∴ 5√2 = \(\frac{p}{q}\), where p and q are some integers and HCF(p, q) = 1 …(1) ⇒5√2q = p ⇒(5√2q)² = p² ⇒ 2(25q²) = p² ⇒ p² is divisible by 2 ⇒ p is divisible by 2 ….(2) Let p = 2m, where m is some integer. ∴5√2q = 2m ⇒(5√2q)² = (2m)² ⇒2(25q²) = 4m² ⇒25q² = 2m² ⇒ q² is divisible by 2 ⇒ q is divisible by 2 ….(3) From (2) and (3) is a common factor of both p and q, which contradicts (1). Hence, our assumption is wrong. Thus, 5√2 is irrational.

Answer»

Let 5√2 is a rational number.

∴ 5√2 = \(\frac{p}{q}\), where p and q are some integers and HCF(p, q) = 1 …(1)

⇒5√2q = p

⇒(5√2q)² = p²

⇒ 2(25q²) = p²

⇒ p² is divisible by 2

⇒ p is divisible by 2 ….(2)

Let p = 2m, where m is some integer.

∴5√2q = 2m

⇒(5√2q)² = (2m)²

⇒2(25q²) = 4m²

⇒25q² = 2m²

⇒ q² is divisible by 2

⇒ q is divisible by 2 ….(3)

From (2) and (3) is a common factor of both p and q, which contradicts (1).

Hence, our assumption is wrong.

Thus, 5√2 is irrational.

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